Is there a finite number of theorems in Mathematics?

Question

I was just thinking, is it possible to prove that certain mathematical field (e.g. linear algebra) has a finite number of theorems? If yes, do all the areas of mathematics have finite number of theorems or are there some areas with infinite number of theorems?

Answer 1

Let $\phi_n$ be the statement $x=x \land \ldots \land x = x$ with $n$ occurrences of the equation $x = x $. Every $\phi_n$ is a theorem of first-order logic, so that gives you an infinite number of distinct theorems in any first-order logic.

To give a better answer to your question, we would need to have a good notion of when two theorems "mean the same thing". That is a difficult (and, in this context) probably unanswerable problem.